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Simplify Rational Expressions

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Simplifying Rational Expressions

A rational expression is a fraction whose numerator and/or denominator are polynomials. To simplify one, you rewrite it in an equivalent form with no common factors left in the numerator and denominator.

1) Factor completely

Look for common factors, difference of squares, trinomials, or grouping. The expression cannot be simplified correctly until all parts are factored as far as possible.

2) Cancel only common factors

If the same factor appears in both the numerator and the denominator, you may cancel it. Cancel factors, not terms. For example, you can cancel \(x-2\) with \(x-2\), but not a term that is added or subtracted unless it is part of a factored group.

3) Rewrite the result

After canceling, multiply what remains in the numerator and denominator. Make sure the final answer is fully simplified.

4) Check for restrictions

Any value that makes the original denominator zero is not allowed. Those restrictions still matter even if the factor was canceled in the simplified form.

Quick check

Your final expression should have no common factor left between numerator and denominator. If possible, verify by multiplying the simplified expression back by the canceled factor to see whether you recover the original expression.

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